Complex Analysis
Qualifying Examination
August 2015
1. Find every complex number 𝑧 for which the infinite series
∞ (
) 2(
)
∑
2015 + 𝑖 𝑛 𝑧 − 2015 𝑛
𝑛=1
converges.
2015 − 𝑖
𝑧 + 2015
2. Determine every complex number 𝑤 that can be written in the form sin(𝑧) for some complex
number 𝑧 having positive imaginary part. In other words, what is the image of the open
upper half-plane under the sine function?
∞
3. Prove that
∫0
(log 𝑥)2
𝜋3
𝑑𝑥
=
.
8
1 + 𝑥2
4. When 𝑛 is an integer, the Bessel function 𝐽𝑛 (𝑧) can be defined to be the coefficient of 𝑡𝑛 in
the Laurent series about the origin of
))
( (
1
1
exp 𝑧 𝑡 −
2
𝑡
(series with respect to the variable 𝑡). Use this definition to show that 𝐽−𝑛 (𝑧) = (−1)𝑛 𝐽𝑛 (𝑧).
5. When the variable 𝑧 is restricted to the first quadrant (where Re 𝑧 > 0 and Im 𝑧 > 0), how
many zeroes does the polynomial 𝑧2015 + 8𝑧12 + 1 have?
6. Suppose 𝑓 is an entire function such that 𝑓 (𝑥 + 0𝑖) is real for every real number 𝑥, and
𝑓 (0 + 𝑦𝑖) is real for every real number 𝑦. Prove the existence of an entire function 𝑔 such
that 𝑓 (𝑧) = 𝑔(𝑧2 ) for every complex number 𝑧.
7. Does there exist a holomorphic function that maps the open unit disk surjectively (but not
injectively) onto the whole complex plane?
8. Determine the group of holomorphic bijections (automorphisms) of { 𝑧 ∈ ℂ ∶ |𝑧| > 1 },
the complement of the closed unit disk.
9. On the punctured plane ℂ ⧵ {0}, can the function 𝑒1∕𝑧 be obtained as the pointwise limit of
a sequence of polynomials in 𝑧?
10. Prove that if 𝑓1 and 𝑓2 are holomorphic functions with no common zero in a region of the
complex plane, then there exist holomorphic functions 𝑔1 and 𝑔2 such that 𝑓1 𝑔1 + 𝑓2 𝑔2 is
identically equal to 1 in the region.
[Exactly 100 years ago, the algebraist J. H. M. Wedderburn proved this proposition by
applying Mittag-Leffler’s theorem.]